Article S-Matrix of Nonlocal Scalar Quantum Field Theory in Basis Functions Representation
Ivan V. Chebotarev 1, Vladislav A. Guskov 1, Stanislav L. Ogarkov 1,2,* and Matthew Bernard 1,*
1 Moscow Institute of Physics and Technology (MIPT), Institutskiy Pereulok 9, 141701 Dolgoprudny, Russia; [email protected] (I.V.C.); [email protected] (V.A.G.) 2 Dukhov Research Institute of Automatics (VNIIA), Sushchevskaya 22, 127055 Moscow, Russia * Correspondence: [email protected] (S.L.O.); [email protected] (M.B.)
Received: 18 December 2018; Accepted: 12 February 2019; Published: 19 February 2019
Abstract: Nonlocal quantum theory of a one-component scalar field in D-dimensional Euclidean spacetime is studied in representations of S-matrix theory for both polynomial and nonpolynomial interaction Lagrangians. The theory is formulated on coupling constant g in the form of an infrared smooth function of argument x for space without boundary. Nonlocality is given by the evolution of a Gaussian propagator for the local free theory with ultraviolet form factors depending on ultraviolet length parameter l. By representation of the S-matrix in terms of abstract functional integral over a primary scalar field, the S form of a grand canonical partition function is found. By expression of S-matrix in terms of the partition function, representation for S in terms of basis functions is obtained. Derivations are given for a discrete case where basis functions are Hermite functions, and for a continuous case where basis functions are trigonometric functions. The obtained expressions for the S-matrix are investigated within the framework of variational principle based on Jensen inequality. Through the latter, the majorant of S (more precisely, of − ln S) is constructed. Equations with separable kernels satisfied by variational function q are found and solved, yielding results for both polynomial theory ϕ4 (with suggestions for ϕ6) and nonpolynomial sine-Gordon theory. A new definition of the S-matrix is proposed to solve additional divergences which arise in application of Jensen inequality for the continuous case. Analytical results are obtained and numerically illustrated, with plots of variational functions q and corresponding majorants for the S-matrices of the theory. For simplicity of numerical calculation, the D = 1 case is considered, and propagator for free theory G is in the form of Gaussian function typically in the Virton–Quark model, although the obtained analytical inferences are not, in principle, limited to these particular choices. Formulation for nonlocal QFT in momentum k space of extra dimensions with subsequent compactification into physical spacetime is discussed, alongside the compactification process.
Keywords: quantum field theory (QFT); scalar QFT; nonlocal QFT; nonpolynomial QFT; Euclidean QFT; S-matrix; form factor; generating functional; abstract functional integral; Gaussian measure; grand canonical partition function; basis functions representation; renormalization group; compactification process
1. Introduction
The timeline of Quantum Field Theory (QFT) offers events that are quite asymmetric to each other. Brilliant triumphs, on the one hand, in explanations and predictions of different processes at low-energy Quantum Electrodynamics (QED), high-energy Quantum Chromodynamics (QCD), Theory of Critical Phenomena, and several other branches of the modern science; but catastrophe, on the other hand, in various attempts to describe high-energy physics in scalar theory and QED levels as well as low-energy QCD physics. Identically, in discoveries of renormalizable field theories,
Particles 2019, 2, 103–139; doi:10.3390/particles2010009 www.mdpi.com/journal/particles Particles 2019, 2 104 the conjecture that only those make sense is opposing to conjecture that there are any and all theories. Nonetheless, robust indeterminacy is fueled by nonspeculative mathematical theory. In non-Abelian Gauge QFT, unifying electromagnetic and weak interactions into a general model, weak interaction processes are consistently described. In another triumph of non-Abelian QFT, the Standard Model (SM), high-energy problem of QED vanishes in chromodynamic reaction channels, because QCD consistently in a high-energy region has the quintessential property of asymptotic freedom. Recently, the discovery of the Higgs boson, the last SM element in the energy domain, where existence is most natural, occurred. Notably, by the remarkable event confirming validity of SM, a quantum-trivial local scalar field quantum theory is, after all, not quantum-trivial if the SM is a sector of a non-Abelian gauge theory; this is analogous to a QED event. Supersymmetric non-Abelian QFTs as well as Integrable QFTs [1–9] in earnest search for gravity quantum theory and naturally complement superstring theory. Under such sophistication for superstring theory, which is almost surely the strongest pick for the fundamental theory of nature, the ultimate truth, it is not impossible to view all QFTs as effective (low-energy) theory given by renormalizable and nonrenormalizable QFTs, respectively. In other words, every field theory is a limit in superstring theory. Hypothetically, bosonic strings are given consideration if tachyon degrees of freedom form a condensate that is consistently separable by physical expressions; if a perfect factorization, not known to date, is found [10,11]. QFT, symmetrical and fundamental, is nevertheless consistent in the framework; superstring theory is field theory. A Euclidean nonlocal QFT with nonpolynomial interaction Lagrangian is robust theory, mathematically rigorous and logically closed [12–18], dual to statistical physics models, encouraging, among other things, further study [19–25] by a statistical physics analogy that in the reverse direction, informs every structure of nonlocality form factors. Under a robust QFT, the existence of nonlocality, be it fundamental or a phenomenon, in the special case of interest of a four-dimensional spacetime, is undeniable by all uncertainties. The question of how analytical continuation in the Minkowski spacetime is arranged is, however, open due to the lack of existence of the no-go theorem [13,14]. At the auspicious moment, nonlocal QFT is really a self-consistent theory to ensure whether observed processes can or cannot be explained. The nonlocal form factor introduced from physical point of view accounts for meaningful physical processes at too small distances, but is an oversight of experiment design. A QFT problem is considered solved if the mathematical apparatus is created for calculus of the S-matrix of the theory, which is the set of all probability amplitudes of possible transitions between the physical system states under consideration. In hadronic interaction (light hadrons) low-energy physics, nonlocal QFT is called nonlocal quark theory; the Virton–Quark model [14,26–30] is considered effective theory for describing quark confinement field due to no additional field, typically a gluon field, required to ensure quark confinement. With first-principle QCD, it is impossible to obtain satisfactory description of low-energy hadronic interactions. Studies use the robust, original hypothesis that quarks do not exist as arbitrary physical particles, but exist only in a virtual quasiparticle state. The Virton field, in the QFT framework, satisfies two conditions: the free state field is identically zero, and the causal Green function, the field propagator, is nonzero. That is, nonobservability (or nonexistence) simply means identical zero free Virtons field; a free Virton does not exist, and Virtons only exist in a virtual state. In the framework of a functional integral, there is nontrivial generating, functional for the theory, in particular for S-matrix of the theory. Further generalizations of nonlocal QFT, and nonlocal quark theory in particular, include the interaction of electromagnetic field with Virtons. Moreover, nonlocal QFT also arises in a functional (nonperturbative, exact) renormalization group (FRG) [31–34]. Ultraviolet form factors are functions of differential operators (in coordinate representation) and they correspond to FRG regulators, which are the regulators of the FRG flow of different generating functionals in QFT and statistical physics. Particles 2019, 2 105
While FRG regulators are not always chosen for entire analytic functions, they are the most preferred for equations of FRG flow with the best analytical properties. Fundamental contributions to formation and development of nonlocal QFT were made, thanks to Gariy Vladimirovich Efimov, in his earlier papers devoted to local QFT with nonpolynomial interaction Lagrangians [13,35,36], and later papers devoted to nonlocal QFT [12,37–39]. His earlier study was developed in parallel but independently in papers of Efim Samoylovich Fradkin [40–42]; hence, the Efimov–Fradkin Theory, which today in its own right is a subject of study by several authors [43–53]. Furthermore, alternatives of the nonlocality were studied by several authors [54–60]. The idea of nonlocality also was developed in a series of papers by J.W. Moffat and co-authors in the early 90s [61–65], as well as in the 2010s [66–68], in particular, in the context of quantum gravity. Since the study of nonlocal theories is motivated, in particular, by attempts to construct the quantum theory of gravity and QFT in curved spacetime, let us briefly discuss papers of J.W. Moffat [66,68] and possible generalizations. In these papers, a modification of the interaction action of graviton with matter (corresponding Feynman vertex) was proposed using nonlocal form factors depending on the Laplace–Beltrami operator, which is a generalization of the usual Laplace operator to the case of curved spacetime. Please note that such a generalization is uniquely possible only in the case of spacetime with constant curvature. If we consider arbitrary spacetime, there is an infinite tower of arguments on which the form factor can depend. These arguments are different combinations of covariant derivatives, curvature scalar, et cetera. However, in this case, we can again consider the form factors that depend only on the Laplace–Beltrami operator, as a special case of a general problem. Proof of unitarity of the S-matrix in nonlocal QFT [69], and causality in nonlocal QFT [70] were done by G.V. Efimov together with Valeriy Alekseevich Alebastrov. Moreover, in what is considered pinnacle [71,72] of set of papers devoted to the nonlocal QFT, G.V. Efimov introduced and investigated the notion of representations of S-matrix of QFT on a discrete lattice of basis functions for both the nonpolynomial [71] and the polynomial [72] theory cases. The G.V. Efimov’s research is described in three detailed monographs [13,14,28] brilliantly, not catastrophically, but also in a review [30]. A remarkable feature of the chosen G.V. Efimov papers [71,72] (see also [14]) is the formulation technique, preserving investigation of strong coupling mode in S-matrix of different QFTs. Moreover, any strong coupling method is as good as gold within QFT, even today. The S-matrix of the nonlocal QFT with nonpolynomial interaction Lagrangian is reducible into grand canonical partition function, the object well known in statistical physics. This implies a paradigm shift, with the new face of results at crossover of two fundamental sciences. As in the Efimov’s papers, S-matrix studied in this paper in Euclidean metric is applicable to finite volume V (QFT in box) study. For a nonlocal theory, basis functions are eigenfunctions of quantum particle in D-dimensional space Schrödinger equation, with multidimensional infinite deep well potential terms, on which further expansions are carried out. Limit of infinite volume is taken at end of derivations; and, following necessary renormalization, final expressions for different physical quantities are obtained. The approach is analog of statistical physics model, where corresponding limit is called thermodynamic limit. However, all summations are replaced by integration, for instance, by Poisson summation, in particular, by Euler–Maclaurin formula. Consciously, drawing attention to the Efimov’s papers [71,72] (see also [14]) again, we zero in on fundamental differences with this paper: We consider QFT in the whole space initially. At same time, on-and-off switching interaction function g is not constant. That is, coupling constant is an infrared smooth function, chosen such that for large values of spatial coordinate x, the interaction Lagrangian vanishes. This approach is similar to fundamental method described in the monograph of Bogoliubov and Shirkov [73,74] and quite nontrivial, since the function g does not change any fundamental property of the phase space system, namely: momentum variable k runs through continuous range of values, and summation over momentum is the initial integral. Moreover, we choose for all expansion basis functions, Hermite basis functions given by eigenfunctions of quantum particle in D-dimensional space Schrödinger equation with multidimensional isotropic oscillator potential. This choice is also Particles 2019, 2 106 similar to what is described in the monograph by Bogoliubov and others [75]. Resulting S-matrix is in representation of basis functions on a discrete lattice of functions. Another next distinct feature of this paper is we do not adopt any basis at all initially. The theory begins with an S-matrix expression, more precisely, the related generating function Z in terms of abstract functional integral over the (scalar) primary field [31,76,77]. By formal algebraic operations, the prior is represented as a grand canonical partition function. For the nonpolynomial theory, the resulting partition function is a series over the interaction constant with a finite radius of convergence. In addition, in principle, it is possible to establish majorizing and minorizing series, the majorant and minorant, as in nonpolynomial Euclidean QFT such as the sine-Gordon model, or in the case of polynomial term modulated by Gaussian function of field [14]. As a result, the S-matrix is studied through statistical physics methods. In the case of the polynomial QFT, resulting partition function is understood in the terms of a formal power series over the interaction constant. The obtained series, in both the polynomial and nonpolynomial cases, is but intermediate step in derivation of expression for the S-matrix in the representation of basis functions. Further derivation for the desired expression for S-matrix of the theory, in representation of basis functions, is done with the expansion of propagator of the free theory G into an infinite series over a separable basis of Hermitian functions from the S-matrix expression in terms of the partition function. Moreover, such decomposition always exists, since propagator of the free theory G is a square-integrable function, the well-known fact of Hilbert space theory. Nonlocal QFT is then formulated with basis functions parameterized by continuous variable k. Practically repeating derivations, the expression is gotten for S-matrix of the theory: the S-matrix in representation of basis functions on a continuous lattice of functions. The advantage of this approach is that in all integrals with respect to momentum variable k, the integration measure dµ is treated as arbitrary variable which is intuitively, mathematically correct since dµ is a formal variable, regardless of its actual value and meaning. The resulting S-matrix, in representation of basis functions on a continuous lattice of functions, is a mathematically defined object for different mathematical analysis techniques, for instance, inequality methods. Jensen inequality, the central inequality of this paper, is used, in particular, to formulate resulting S-matrix (strictly speaking, − ln S) in terms of variational principle: constructing majorant of corresponding lattice integral in terms of variational function q, and minimizing the resulting majorant with respect to q; a well-known technique, for instance, in polaron problem in terms of functional integral (polaronics [78]). However, we note that in continuous case, upon applying Jensen inequality, additional divergences arise due to structure of Gaussian integral measure Dσ (not to be mixed with dµ) on continuous lattice of variables; but, plausibility of a representation on continuous lattice of functions is not negated. Moreover, the presented divergence problem, in principle, is solved by the FRG method: Finding the FRG flow regulator RΛ containing measure dµ as its argument, and carrying out formal derivations such that renormalization (rescaling) of strictly defined “building blocks of theory” in terms of the physical equivalents is done in final derivation; an approach which is analogous to the Wilson RG, consisting of mainly two steps, decimation and rescaling [31]. However, the approach leaves several questions unanswered: What is then the connection of original S-matrix of the theory to representation in basis functions on continuous lattice of functions? Is it possible to redefine an original expression for S-matrix by rescaling the same strictly defined building blocks of the theory? In addition, in what sense can the Jensen inequality be further understood? The FRG approach reminds us of “Cheshire Cat”, the mischievous grin in suggestion of how the S-matrix of the theory is to be defined from the very beginning. As a result, we then give an alternative formulation for the S-matrix of the theory where results on a discrete or continuous lattice of basis functions are consistent in both polynomial and nonpolynomial QFTs, with no question that the mathematical theory is both rigorous and closed. Particles 2019, 2 107
In summary, the motivation and methodology of this paper have been described. In the next section, the methodology is implemented in framework of generalized theory. In section three, the focus is on polynomial ϕ4 theory in D-dimensional spacetime. The variational principle is restricted to the case satisfying separable kernels of the resulting lattice and integral formulae; this approach increases the value of the majorant, which is consistent with the framework. In latter part of the section, the case of polynomial ϕ6 theory in D-dimensional spacetime is discussed. In section four, nonpolynomial sine-Gordon theory in D-dimensional spacetime are studied; separable kernels of the lattice and integral equations are considered. In section five, numerical results are presented for prior obtained analytical expressions. To simplify computations, the case D = 1 is observed, although analytical inferences obtained do not only apply to D = 1, in principle. Further simplifying computations, the Gaussian function is chosen as propagator for the free theory G, since such model is close to reality: the propagator G in the form of Gaussian function, in particular, is typical in the Virton-Quark model, which is the generally accepted model for describing light hadrons low-energy physics [14,26–30]. Moreover, with the Gaussian choice of propagator, final expressions only arrive in simple, transparent form. In latter part of section five, we give an original proposal, consisting of the main of the reinterpreting the concept of nonlocal QFT. We propose considering the theory in internal space with extra dimensions and subsequent compactification into physical four-dimensional spacetime. This not only changes the meaning of nonlocal interaction ultraviolet length parameter l, the needle of pivot for ratio of ultraviolet and infrared parameters in the theory, it also changes analytical properties of the Green functions, scattering amplitudes, and form factors, in terms of physical variables obtained following compactification from internal space with extra dimensions; yet, method compactification is determined by process [10,11,79]. In other words, in section five, we propose to change the concept of connections in nonlocal QFT. In the conclusion, a lasting discussion on all obtained results and inferences is left. In conclusion, let us make one remark. All calculations in the paper are performed in Euclidean spacetime. The same logic is valid in standard loop calculations in the local QFT. This technique allows to calculate various Feynman diagrams in the simplest way. In all final results Wick rotation must be made. At the same time, calculations in the Euclidean spacetime are a separate independent problem. The results of such calculations can be useful for quantum field models of statistical physics, in particular, in the Theory of Critical Phenomena, Quantum Theory of Magnetism, and Plasma Physics. The construction of analytical continuations for the results of this paper is the subject of a separate publication.
2. General Theory
2.1. Derivation of the S-Matrix in Terms of the Grand Canonical Partition Function In this subsection, we obtain an expression for the S-matrix in terms of the grand canonical partition function, which in turn is to be expressed in terms of the abstract functional integral over the primary fields of the theory. Let the generating functional Z be given by an abstract functional integral for the free (Gaussian) theory, similar to Gaussian QFT with classical Gaussian action [9]. That is, Z is given by [31,76–78]: Z 1 −S0[ϕ]+(j|ϕ) (j|Gˆ|j) Z[j] = D[ϕ]e = Z0e 2 (1)
1 ˆ ˆ ˆ −1 0 00 where S0[ϕ] = 2 (ϕ|L|ϕ) is action of the free theory, L = G is inverse propagator, j = j + ij ∈ C is source, which in general is a complex field, (j|ϕ) = R dDzj(z)ϕ(z) is scalar product of the source and primary field (the definition is given for real-valued j and ϕ configurations), and the quadratic form of the source is generally given by integration over x and y: (j|Gˆ|j) = R dDx R dDyj(x)G(x − y)j(y). The generating functional Z of the theory with interaction is written in the form [31,76–78]: Z Z[g, j] = D[ϕ]e−S0[ϕ]−S1[g,ϕ]+(j|ϕ) (2) Particles 2019, 2 108
where S1[g, ϕ] is interaction action, the action responsible for system interaction, given by [13,14]:
+∞ Z Z Z dλ Z S [g, ϕ] = dDxg(x)U[ϕ(x)] = dDxg(x) U˜ (λ)eiλϕ(x) ≡ dΓeiλϕ(x) (3) 1 2π −∞ such that Γ is a notation for a point in the system phase space, which is the product of spaces λ and x; the class of the function U(ϕ) does not play a role, up to certain point, but the Fourier transform U˜ (λ) of the interaction Lagrangian is a distribution, in general, for instance, if U(ϕ) is a polynomial in the primary field ϕ. Function g(x) is the infrared coupling “constant”, switching on-off the interaction. The generating functional Z in (2) expands into functional Taylor series over coupling constant g:
( ) n Z ∞ n n Z i ∑ λa ϕ(xa) −S [ϕ]+(j|ϕ) (−1) Z[g, j] = D[ϕ]e 0 ∑ ∏ dΓa e a=1 . (4) n=0 n! a=1
Therefore, the following representation is valid: Z D (D) iλa ϕ(xa) = d ziλa ϕ(z)δ (z − xa) = iλaδ•xa ϕ where scalar product is defined in terms of integral; and, notation X• means variable X without explicit argument, for instance, the result of integration which is independent of the integrand variable. By the resulting representation, expression (4) is given by:
n ∞ n ( n ) (−1) Z Z −S0[ϕ]+ j+i ∑ λaδ•xa ϕ Z[g, j] = ∑ ∏ dΓa D[ϕ]e a=1 n=0 n! a=1 ( ) n n ∞ n n Z 1 j+i Gˆ j+i (−1) 2 ∑ λaδ•xa ∑ λbδ•xb = = = ∑ ∏ dΓa Z0e a 1 b 1 n=0 n! a=1 n n ∞ n ( n ) − 1 ( − )+ ˆ ( ) (− ) Z 2 ∑ λaλbG xa xb i ∑ λaGj xa 1 a b= a=1 = Z[j] ∑ ∏ dΓa e , 1 . n=0 n! a=1
The result of action of the operator Gˆ on the source j is given by Z Gjˆ (x) ≡ dDzG(z − x)j(z) ≡ ϕ¯(x).
We now introduce a central object of QFT: the S-matrix of the theory, given in terms of functional Z by an explicit expression (similar to [13,14]):
n −1 ∞ n ( n ) − 1 λ λ G(x −x ) Z[g, j = Gˆ ϕ¯] (−1) Z 2 ∑ a b a b S[g, ϕ¯] ≡ = dΓ eiλa ϕ¯(xa) e a,b=1 . (5) ˆ −1 ∑ ∏ a Z[j = G ϕ¯] n=0 n! a=1
In particular, by above expression (5), the S-matrix of the theory is given in terms of the grand canonical partition function. If Fourier transform of the interaction Lagrangian U˜ (λ) is classical, for instance, quadratically integrable function, the resulting partition function is a series with finite radius of convergence, and majorizing and minorizing series, the majorant and the minorant, respectively, are obtainable, as in nonpolynomial Euclidean QFT, for instance, in sine-Gordon model, or when the term polynomial in the field is modulated by the Gaussian function. Consequently, the S-matrix of the theory is studied by statistical physics method. If original interaction Lagrangian is polynomial in the primary field ϕ, the Fourier transform of U˜ (λ) is a distribution; and, expression (5) is a formal power series with respect to the interaction constant, on an expansion of the propagator G into an infinite series over a separable basis. In particular, Particles 2019, 2 109 such decomposition always exists, since the propagator G is square-integrable function, a fact well known from theory of Hilbert spaces. In the next subsection, we discuss the propagator splitting. In conclusion, in this subsection, we pinpoint what is the meaning of the term “nonlocal QFT” in context of this paper. A priori, such term is formulated as a theory with standard Gaussian propagator Gˆloc of local QFT, and nonlocality is implied by the argument of function U in (3) in a composite operator form Kˆ ϕ. However, in this paper by change of variables Kˆ ϕ → ϕ in the functional integral (4), the a priori theory is reduced to the form of a theory with local interaction U(ϕ), 2 2 and whose propagator Gˆ equals Kˆ Gˆloc, with the operator Kˆ playing the role of ultraviolet form factor. As a result, the propagator Gˆ is closely heeded everywhere in this paper.
2.2. Propagator Splitting Central to the contextual theory of nonlocal QFT is propagator splitting, the expansion of the propagator of the free theory G into an infinite series over a separable basis; in the following steps, G is first represented, similar to [14,71,72], in the form Z G(x − y) = dDzD(x − z)D(z − y). (6)